there is a mystery at the heart of our
universe a puzzle that so far no one has
been able to solve it can be weird
welcome to the world if we can solve
this mystery it will have profound
consequences for all of us and that
mystery is why mathematical rules and
patterns seem to infiltrate pretty much
everything in the world around us
many people have in fact described maths
as the underlying language of the
universe but how did it get there even
after thousands of years this question
causes controversy we still can't agree
on what maths actually is or where it
comes from is it something that's
invented like a language or is it
something that we have merely discovered
I think discovered invented it's both I
have no idea why does any of this matter
well maths underpins just about
everything in our modern world from
computers and mobile phones to our
understanding of human biology and our
place in the universe my name is Hanna
Frey and I'm a mathematician in this
series I will explore how the greatest
thinkers in history have tried to
explain the origins of maths
extraordinary power
human dissipation I'm going to look at
how in ancient times our ancestors
thought maths was a gift from the gods
how in the 17th and 18th centuries we
invented new mathematical systems and
used them to create the scientific and
industrial revolutions and I'll reveal
how in the 20th and 21st centuries
radical new theories are forcing us to
question once again everything we
thought we knew about maths and the
universe the unexpected should be
expected because why would reality down
there bear any resemblance to reality up
here
[Music]
in this episode I go back to the time of
the ancient Greeks to find out where our
fascination with numbers started you
know I think I can hear the neighborhood
cats screeching and reveal why we're now
looking for maths deep inside our brains
[Music]
our world is full of maths often in
unusual places like this rollercoaster
[Music]
this ride wouldn't be possible without
physics and engineering and at the heart
of all of that science it's math master
oh my god
[Music]
it's sobering to think how much we
entrust our personal safety to matter
without even realizing it
[Music]
my rush of adrenaline relies on
someone's calculations of kinetic energy
momentum tensor strengths coefficients
of friction and much much more do you
make me do
[Music]
so pretty bluntly the modern world
wouldn't exist without mathematics he is
hiding behind almost everything that's
around us and subtly imprinting almost
everything that we now do and yet it's
invisible it's intangible so where does
mathematics come from where des numbers
live
[Music]
it's a question that goes to the very
heart of our world we often think about
numbers as something tied to objects
like the number of fingers on one hand
or the number of petals on a flower this
flower has got eight petals if I take
three away then it will be left with
just five I don't look a lot less pretty
the petals are gone but the number three
still exists the idea of three or any
other number for that matter is still
out there even if we destroy the
physical object but you can't say that
about everything
if pencils had never been invented then
the idea of a pencil wouldn't exist but
the idea of numbers would still exist in
every culture around the world we all
agree on what the concept of Faunus is
like and it doesn't matter whether it's
called for cat fur fear or even what the
symbol looks like
[Music]
with numbers I can destroy the physical
object burn it to a crisp but I can't
destroy the idea of numbers so here's
the question I want to answer is it
invented or discovered is there some
magical parallel world somewhere where
all mathematics lives a place where you
have fundamental truths that help us to
understand the rules of science helping
us put man on the moon and to study the
tiniest particles of the universe or is
maths all in our minds is it just a
figment of our imagination and intellect
where the maths is invented or
discovered is something we can't agree
on it's just too extraordinary to think
that the mathematical truths and
everything is sort of product entirely
of our conventions and the human mind
and that's I don't think we're that
inventive it sometimes feels like
mathematics is discovered especially
when the work is going really well and
it feels like the equations are driving
you forwards but then you take a step
back and you realize that it's a human
brain that's imposing these ideas these
patterns on the world and from that
perspective it feels like mathematics is
something that comes from us the number
five is called FEM and Swedish my
mother-tongue that part we invent the
baggage the description the language of
mathematics but the structure itself
like the number five and the fact that
it's two plus three that's the part that
we discover
there's virtually no part of our
existence that isn't touched by maths so
if it is discovered part of the fabric
of the universe how can we unlock its
secrets and if it's invented and all in
our heads
how far can our inventive brains take us
I want to start with the discovered camp
those who say maths is all around us
you just need to know where to look of
all of the structures that you get in
nature only think one of the most
beautiful is the nautilus shell so
there's a little creature that lives
inside a and creates all these shapes
and hops from one chamber to the next as
it grows and this shell is just
incredibly intricate
and you might wonder how something so
small can create something quite so
remarkable but actually there is a
hidden pattern in here that you can
start to see when you measure these
chambers so that one is coming out at
fourteen point five
and this one on the same axis is 46.7
I'm measuring how wide the shell would
have been as the Nautilus grew I pick an
angle and measure the inner chamber and
then a second measurement to the outer
rim ninety nine point I do this three
times for three different angles until I
have three sets of numbers when you look
at them I mean they look pretty random
right looks like there's no connection
between them at all but looks can be
deceptive because if you take each of
these pairs of numbers and divide one by
the other a very clear pattern starts to
emerge so here if we do this number
divided by this we get 3.2 - this number
divided by this one gives 3.25 I think
sorry I meant arithmetic isn't great I
think this number divided by this number
then gives 3.24 and suddenly the same
number starts to appear around about 3.2
ish doesn't matter where on the shell
you measure the ratio of the width of
these chambers ends up being pretty much
constant throughout the show I've got it
right to one decimal place
bad that will be down to my measuring
skills rather than the Nautilus what all
of this means is that the Nautilus is
growing its shell at a constant rate so
every time it does a complete turn it
ends up sitting in a chamber that is
around about three point two times the
width of the turn before and by
repeating this very very simple
mathematical rule it can create this
beautifully intricate spiraled shell
clever author lists the Nautilus isn't
the only living thing that has a
mathematical pattern hidden inside it if
you've ever counted the petals on a
flower you might have noticed something
unusual some have three petals some five
some eight some 13 but rarely any of the
numbers in between these numbers crop up
time and time again may seem random but
they're all part of what's called the
Fibonacci sequence you start with the
numbers 1 & 1 and from that point do you
keep adding up at the last two numbers
so one on one is - 1 + 2 is 3 2 + 3 is 5
and so on
when looking at the number of petals in
a flower these numbers from the
Fibonacci sequence keep appearing but
that's just the start if you look at the
head of a sunflower you'll see the seeds
are arranged in a spiraling pattern
count the number of spirals in one
direction and you will often find a
Fibonacci number then if you count the
spirals going in the opposite direction
you'll hit upon an adjacent Fibonacci
number why do plants do this well it
turns out that this is the best way for
the flower to space out its seeds so
they don't get damaged we find these
spirals so intriguing we've worked hard
to unlock their secrets we've gotten
very good at copying the patterns that
we find in nature and using them to
create things of great beauty like this
majestic stair
[Music]
simple glorious mathematical rules found
hidden in nature doesn't seem to me like
a coincidence
these mathematical patterns once you
spot them do feel discovered it's as if
the maths is already out there
I'm just waiting for you to find it
[Music]
this fascination for finding hidden
mathematical patterns is nothing new
go back over 2,000 years to the time of
the ancient Greeks and you will find the
philosopher Pythagoras and his followers
were just as enthralled by the patterns
they discovered the pythagorean's were
obsessed with numbers they were people
who believed that numbers were a gift
from God and part of their fascination
might have been thanks to their
experiments with music the pythagorean's
discovered patterns that linked the
sound of beautiful music to the length
of a vibrating string this they believed
was no accident but a window into God's
worlds that had been gifted to the
pythagorean's mathematician and musician
Ben sparks is fascinated by this age-old
relationship between music and maths
[Music]
[Applause]
thank you for joining us on our with
your beautiful cello there okay then you
are you're gonna have to explain this to
me
where does the math come in in making
this instrument sound nice the wobbling
is what's giving you the sound and if
you make the string wobble you hear a
sound so maybe you could play your D
string
Oh sounds lovely sounds lovely doesn't
it and what they also notice is another
no really related to that which is
making if you make it wobble twice as
fast and to do that you can make the
string half the length okay
so you're putting your thing here why I
guess pretty much is that not hard way
along okay what's weird about these two
noses they they sound kind of the same
but they're definitely different and
this is where the Greeks notice they we
call it an octave but if you play them
together does it sound nice
delightfully Pleasant in the octave the
length of the vibrating string creates a
relationship or ratio of two to one so
that's if you chop it in half
are there other fractions that make it
sound nice well exactly what the Greeks
were thinking is like what can we find
other notes that sound even nicer to get
a more interesting together can you play
us this is what they call a perfect
fifth
what happens when you play those two
together there
very pleasant the first we're about to
launch into a jig yeah in a perfect
fifth the ratio of the vibrating string
is three to two the high note
is two thirds of the length of the low
note what happens when you play a note
that isn't one of these neat fractions
when notes aren't in these nice simple
ratios we tend to notice that even if
we're not aware of the mathematics right
I mean can you play a sort of really
horrible harmony together maybe like a
semitone apart when the strings are not
in a simple ratio the harmony sounds
distinctly unpleasant the Greeks were
obsessed with having simple ratios
describing the notes so they get nice
harmonious noises how does this work for
other instruments I mean this is very
clear you've got this this sort of
string here but what about and I don't
know like the human voice right well
every noisy over here is things wobbling
somehow whether it's your vocal cords or
a string or my vocal cords
really have you never used your vocal
cords for a bit of music can we try oh
I'm such a fine singer please don't make
me do this
okay you got your earplugs in yeah let's
try can you picture some note I mean
there's something nice and low if you
just do it to LA then now I've got a
choice just like the cello it's the
length of mine and Ben's vocal chords
that's changing the pitch of these notes
so that was me singing a perfect fifth
carats if I know love you know I think I
can hear the neighborhood cats
screeching so I think that's enough of
that
these patterns convinced the ancient
Greeks that they'd been gifted a glimpse
into this godly young
why else would these patterns exist
[Music]
Pythagoras and his followers were in
little doubt the maths was just as real
as the music was and it was even neater
and more elegant than anything the human
mind could conceive of the pythagorean's
were by no means the first people to use
some form of maths there's some evidence
that marks found cut into bones from the
Upper Paleolithic era 37,000 years ago
we're tally marks used for counting but
it was the pythagorean's who were the
first to look for patterns
it does feel to me as if maths is all
around us and something we discover a
fundamental part of the world we live in
and yet somehow very strangely separate
from it
trying to make sense of this apparent
paradox is at the heart of this battle
about where maths really lives the
philosopher Plato is one of the most
important figures of the ancient Greek
world but what he said about the origins
of maths is still the basis for what
many mathematicians believe today
he was fascinated by the geometric
shapes that could be produced by
following the rules of mathematics rules
that he believed came from God when I
try and draw a circle really really
carefully takes me back to my school
days this now not doing bad that's
pretty good but if you look really
closely it's just not quite perfect to
the circle but I'm not gonna beat myself
up about this because even if I had
access to the most accurate computer in
the world the circle that it would draw
still wouldn't be perfect
zoom in close enough and any physical
circle will have bumps and imperfections
that's because according to Plato
flawless circles don't exist in the real
world he believed the perfect circle
lives in a divine world of perfect
shapes a kind of mathematical heaven
where all of maths can be found but only
if you're a true believer
he was convinced that everything in the
cosmos could be represented by five
solid objects known as the platonic
solids
say the earth was the rock solid cue
fire was the very pointy tetrahedron and
then with eight triangular sides air was
the octahedron while the icosahedron
there was twenty triangular sides
represented water the Lance brittonic
Sollers the dodecahedron this one was
supposed to encapsulate the entire
universe it's the whole universe sitting
any more hands there it's kind of a neat
idea there's something special about the
Platonic solids they're the only objects
where every side is the same shape and
there are only five try as you might you
will never find another object with
these unique mathematical qualities all
of these shapes Plato believed existed
in a world of perfect shapes beyond the
reach of us mere mortals a place we call
the Platonic world I know that these
ideas might seem like they're a bit
bonkers but there are actually quite a
few people who believe them and those
people come across as though they're
sane my third favorite favorite
mathematical structure the octahedron
AHA the Platonic solids I presume it's a
very heated
I love dodecahedra I have a misspent
youth making models of polyhedra oh my
goodness these are the Platonic solids
Oh guys
okay very beautiful you know at 67 mrs.
Christmas can I keep these two please
these platonic solids to me are a great
example of how mathematics is discovered
rather than invented when ancient Greeks
discovered that this one existed they
were free to invent the name of it they
called it the dodecahedron but the pure
double key he drew himself it was always
out there to be discovered I have this
kind of platonic view that there are
triangles out there there are numbers
there are these circles that I'm seeking
to understand so for me they feel like
quite tangible things they're all part
of this mathematical landscape that I'm
exploring but not everyone believes in
this platonic world of mathematical
truths I think that the platonic world
is in the human head it's a figment of
imaginations I get that that there are
people who really buy into this other
realm of reality and especially if your
days and nights are spent thinking about
and investigating and researching this
realm that doesn't mean that it's real
Plato would have strongly disagreed he
encouraged us to believe in this other
world where all of maths could be found
and not to be fooled into thinking the
world around us is all there is what we
perceive as reality he cautioned is no
more than shadows cast on the walls of a
cave Plato had a very lively of quite
dark imagination to explain what he
meant he came up with an analogy of a
group of humans locked in a cave these
people would have been imprisoned since
childhood and they were shackled by
their necks and their legs entrapped
staring a blank wall directly in front
of them
in his mind's eye Plato pictured a fire
burning high above the prisoners heads
but they have no idea it's there on top
of the wall is a path along which all
manner of people and objects are
traveling but the only thing the
prisoners can see of them is the shadows
they cast down the wall those shadows
are the prisoners reality according to
Plato we are no different to the
prisoners in the cave who mistake the
shadows for reality if Plato is right
what does this mean for you and me is
what we think of as reality and maths
just an illusion are we living in
Plato's cave and and just see see
shadows it is not impossible that that
is the case you know we are maybe just
all we're just some simulation in some
world of some more intelligent being
this is all possible I mean if you think
that there's some world of mathematical
objects it's different from ours it's
not the physical world we live in but
that doesn't make the physical world any
less real so I don't think that there's
anything to me in philosophy of maths
that would force you to think that our
world is an illusion of any kind our
senses evolved really for one purpose
survival but survival and the true
nature of reality are two different
subjects so the fact that we have been
able to survive by thinking about the
world one way does not in any way say
that that way of thinking about the
world is truly what's happening out
there
over 2,000 years ago Plato took the
geometry of shapes as evidence of God's
influence ideas that were limited to the
senses and imagination today geometry is
at the cutting edge of science new
technologies have allowed us to look at
the world beyond our senses and once
again it seems the natural world really
is written in the language of maths
this is a model of a virus
straight away you notice it's geometric
shape Plato would have recognized this
shape as one of the Platonic solids if
there's one person who understands
geometry it's a mathematician ride into
a rock is a professor of mathematics at
the University of York she's trying to
work out how viruses use maths to form
their geometric shapes if you know that
you can find a way to stop them that's
why ridin and her colleagues have
designed a computer simulation that puts
the mathematician at the heart of the
virus but we try to understand is how
this virus forms and we taught her to do
that we will create the illusion of
being inside of the virus in the
position where the genetic material
normally is Ryden has discovered that
the virus harness is the power of maths
to build its shell in the quickest and
most efficient way possible armed with
this knowledge she's trying to find a
way to stop the viruses such as
hepatitis B and even the common cold
from developing in the first place once
you understand how this mechanism works
you can turn tables on viruses and
actually prevent that process
that is what makes this research so
exciting if you know the mathematics of
how the virus forms its shell you can
work out the way to disrupt it no shell
no virus no infection
today mathematicians like rydym are
joining the front line in the fight
against disease
[Music]
far beyond the realm of human senses it
really does seem like the universe
somehow knows maths it really is amazing
how often these patterns seem to crop up
there in plants there in marine life
there even in viruses there really is an
awful lot that we can explore and
exploit using the mathematics that we
have it does lend weight to the idea
that there is some natural order
underpinning the world around us so far
it does feel like the idea that maths is
discovered is leading the charge
but perhaps we've been looking for
patterns in the wrong places if it's all
in our heads then the brain feels like a
good place to look is there evidence in
there of maths being an invention of the
human mind I've got a real treat in
store for me today I am heading over to
UCL the University of Iowa cap where
some colleagues are going to scan my
brain and see which bits of it are
working whenever I do mathematics
neuroscientist professor Fred dick is
going to place me inside an fMRI scanner
he'll measure my brain activity by
tracking where the blood flows when I'm
answering questions ranging from
language to maths if my brain treats the
mathematical problems in the same way as
any other problem then it suggests
there's nothing special about maths it's
the same as any other language a clue
perhaps that it's an invention
I'll have ten seconds to think about
each question I don't need to answer out
loud I just have to work out the answer
in my head okay hello how is that well
we didn't want you to relax in there
really I've answered all the questions
to the best of my ability after a few
hours of processing professor Sophie
Scott has my results it's not my brain
that's your brain let me make sure I
understand what I'm seeing it okay so
this is like you've cut my brain in half
yes and I've got the left hand side
there yes and the right hand size is
that so it's like you've chopped my head
down the middle and then split it out
exactly so what you can see here Hannah
is the pattern of activity in your brain
and you're hearing the straightfoward
language and here you can see in the
left hemisphere very classic language
areas activated the bright yellow areas
are where there's increased blood flow
an indication that the neurons in the
left hand side of my brain are working
harder
this is a side of the brain that we know
is linked to language compare that to
the right hand side of my brain where
there's hardly any yellow areas which
means there's far less activity taking
place so can we see maths please Oh hold
on this time when I'm thinking about
maths
there are yellow areas in the right-hand
side of my brain this is very different
to the lack of activity seen when I was
thinking about language these scans
reveal there seems to be a place in our
brains where maths lives what we're
definitely able to say is this is not
just the meanings of the words that you
were reading we're not just looking at
you thinking about the meaning of words
you're seeing something that does seem
to be qualitatively different for the
maths maths is real this is real at
least in my head
at a what really struck me about that
conversation and safe it is then it's
that it doesn't matter whether you're
doing two plus two equals four or
whether you're answering these much
higher level math questions it's the
same bit of your brain that's doing the
grunt work it's not the same thing that
does words or language you're seeing
these problems and you're manipulating
them in your mind research with similar
experiments shows it's broadly the same
for all of us in your brain and mine
there is a specific place where we do
maths
but this doesn't prove that maths is
something we discover it could still be
an invention just one that we learn at
school to get to the bottom of this
question I need some volunteers who've
never had a math lesson in their lives
[Music]
dr. Sam wass is an experimental
psychologist at the University of East
London helping him with some experiments
are six months old IRA and Leo who's
just under a year to begin with each
child is placed in a room where they're
shown a series of images Sam uses a
battery of tests to analyze how they
react to different situations
the first experiment uses eye tracking
technology to see how the baby follows
the movement of a piggy puppet is that
good so here we can see a feed out of
what the child is looking at and those
two red dots are where the baby is
looking what we're presenting is a
puppet that jumps up and then disappears
and it jumps up and disappears two times
in a row and then it stops we present
the same sequence again and again and as
the baby watches it again and again
they're looking times the amount of
attention that they're paying to the
screen diminishes and that tells us that
the Trotters to learn this sequence now
instead of popping up twice as expected
the puppet appears three times does the
child notice the difference between the
tunas of it popping up twice in a row
and the three nests of it popping up
three times and if it does then that
tells us that the child understands the
difference between two and three these
tests reveal that the child is surprised
when the puppet appears more often when
larger scale experiments were carried
out by researchers in the US the results
are
did the infants do have a sense of
quantity so this research is really
important because it's suggested that
even infants as young as five months old
can do the basics of addition and
subtraction they know the difference
between one plus one equals two and one
plus one equals one which is an
incorrect conclusion and that there was
a really really strong provocative
finding yeah this idea that the concept
of mathematics and the basics of
mathematics rules might be hardwired my
DNA in our genetic code this research
isn't conclusive but it does suggest we
all come pre-programmed to do maths some
argue that we evolved this maths part of
our brains to discover the world of
mathematical truths the evidence for
maths being discovered is compelling
we found patterns in nature the latest
technology has uncovered startling
patterns in viruses and scans reveal
there's a part of our brains where maths
lives
but this question is too important to
leave the evidence here and move on if
it is discovered if it lives in this
other world
can we trust what it's telling us how do
we know that our idea of numbers is
right how do we know someone isn't just
gonna come along at some point and say
well actually you've got that completely
wrong and one plus one doesn't equal two
after all how do we know we can rely on
the maths that we take for granted
[Music]
what you need to be sure of is your
foundations if they're shaky then all of
your carefully constructed ideas come
crashing down and there was one
mathematician who understood this only
too well his name was Euclid around 300
BC in Alexandria he wrote one of the
most famous and important books of all
time the elements he was trying to go
right back to the beginning to find the
smallest elements on which you can build
the vast gigantic structure of
mathematics if you have a little flip
through you can see the kind of things
that you cleared was considering so here
it says that you can draw a straight
line between any two points which it's
blindingly obvious and here it says that
all right angles are the same and these
are quite simple concepts but I think
that they really illustrate just how
exhaustive Euclid had to be to build the
foundations for what was to come
he took statements like these which
mathematicians assumed were true and put
them to the test
he then set out to prove a whole host of
other theories based on these
fundamental building blocks this was
really the first time that someone had
written down formal proofs for
mathematical assumptions now
mathematical proof isn't like scientific
proof or proof in a court of law there's
no room for reasonable doubt here
instead if something is true
mathematically once then it is true
forever and that is why this book is so
important it's the reason why Euclid's
elements is still relevant today every
page within it is as true now as it ever
was
and from that point of view it really
does feel like we're tapping into a
world that already exists one that is
just out there waiting to be discovered
unless of course you throw a spanner in
the works change the language of maths
and invent a better way of doing things
suddenly this rock-solid world of
god-given truths might feel decidedly
shaky one thing we know about languages
is that they never stand still they're
constantly evolving to meet the
challenges of a changing world
forty-nine wien let's go for another
tricky one for centuries the language of
mass was thought to be fixed and
unchangeable that is until something was
found to be missing most exactly is zero
zero means nothing if you've got zero
flowers you've got no flowers and if you
guys zero stop zero or something you've
got nothing so you can't really do
anything with the zero I don't really
use it when I'm counting in numbers
before the seventh century neither did
anyone else though people have always
understood the concept of having nothing
the concept of zero is relatively new
we had numbers and could count but zero
didn't exist if you think about it for
long enough zero is actually quite a
strange concept it's almost as though
the absence of anything becomes
something is it just a number or only
dear and how can something with no value
have quite so much power it's not
exactly clear who first thought of zero
it might have originated in China or
India what we do know is that zero
arrived in Europe from the Middle East
at about the same time as the Christian
Crusades against Islam when ideas coming
out of the Arab world were often met
with suspicions the West already had a
numerical system Roman numerals they did
the job but were a bit unwieldy for
example the number 1958 is written as
mcm-l VIII and no matter where you place
say the letter C it will always
represent the number 100 it was good for
its time but times change and a better
system was needed zero was different
where you place zero could change the
values of the numbers around it think of
the difference between eleven and a
hundred and one
although the concept of zero might have
been created elsewhere it was in India
that zero started to be accepted as a
proper number
[Music]
this is a page from the Indian back
Shawnee manuscripts from around AD 225
which shows the dots above the
characters representing zero this is the
earliest known use of the symbol zero
that we know today for almost a thousand
years Indian mathematicians worked
happily with zero while their Western
counterparts piled on with the Roman
numerals
that was until Italian mathematician
Fibonacci recognized its potential now
he'd been educated in North Africa
so he'd seen this number system working
firsthand zero is a placeholder
signifying the absence of a value a zero
is also a number in its own right it
allowed you to write down numbers and
manipulate them much more quickly and
easily than Roman numerals realizing all
this Fibonacci champions the new number
and brought it to the attention of
Western Europe
zero wasn't something that we discovered
it so much as something that was created
as part of a new language to describe
numbers and that's not to say isn't
useful the whole of modern technology is
literally built on ones and zeros but
suddenly maths feels like something
we've come up with something we've
invented we needed a more user-friendly
numerical system so someone came up with
the clever idea of zero not a gift from
the gods but a smart way to make
counting more convenient this is
intriguing evidence that maths might be
invented after all a product of our
intellect and imagination once the idea
of zero had been widely accepted
mathematicians could relax all
conceivable numbers lay out on a single
line with no holes and no gaps to speak
of over here you have the positive
numbers one sheep two donkeys the kind
of stuff you find in real life and in
the other direction all the negative
numbers it's a bit trickier to imagine
what negative one sheep looks like the
number line stretches out in both
directions all the way to infinity and
zero sits proudly in the middle
everything was well in numberland or was
it this is where it all starts to get a
bit strange because there are some
numbers that are simply weird there are
some fundamental rules of maths that you
learned at school two times two equals
four
three times three equals nine a positive
number multiplied by itself equals
another positive number nothing
controversial so far
curiously a negative number multiplied
by itself also gives a positive number
why is that well this is not a math
lecture so let's just accept it as a
fact and move on in fact if you take any
number and multiply it by itself square
it then the answer is always going to be
positive if plus two squared gives me
plus four and minus two squared gives me
plus four what do I have to square to
get minus four but it's a question
without an answer there is no number
that when multiplied by itself gives a
negative answer that is unless you
invent your own meat I a number we
simply made up
[Music]
not everyone was keying it became known
as an imaginary number a deliberately
chosen derogatory term to scoff at its
existence it turns out AI is really
useful especially when it comes to
simplifying problems with things like
electricity or wireless technologies
things that otherwise would seem
impossible to solve essentially if
you're working with waves you will use I
this imaginary number broke all the
rules it didn't come from this world of
ethereal numbers it wasn't god-given it
was very definitely invented if you can
have one imaginary number why can't you
have two or three or infinitely many of
them why can't you have negative
imaginary numbers as well why can't in
fact imaginary numbers have their very
own number line exactly the same as the
real one just on a different axis the
number line isn't a single line at all
numbers are two-dimensional
you might think this all sounds are
beached airy-fairy the imaginary numbers
that we just made up but if you've ever
flown in an aircraft
you've already trucks they're your life
to these strange numbers airports air
traffic controllers here rely on radar
to keep everything moving safely and
quickly once you can busy definitely
need Raider so the busier the tower
the busier the operation you need radar
radar works by sending out radio waves
and examining that part of the signal
that's reflected back the complex
equations that allow us to filter out
the correct signal from other
conflicting frequencies is heavily
dependent on imaginary numbers in this
case separating out moving objects like
planes from flocks of birds or
stationary objects imaginary numbers are
a very efficient tool to be able to
manipulate radio waves imaginary numbers
are fundamental develop eration
imaginary numbers allow us to track
planes in real time without them we
never would have been able to use radar
in our skies
did you know folks father nice try dad
they just told you to text you a Papa
[Music]
when I started this investigation going
back to the time of the ancient Greeks
it did seem like maths could only be
discovered there were too many
coincidences too many mathematical
patterns popping up all over the place
but if we can invent the rules creates
new numbers and they work then
perhaps I've got it wrong maybe maths is
invented after all the concept of zero
or negative numbers or complex numbers
or imaginary numbers they caused great
consternation to the cultures that first
invented or encountered them there are
some conjectures that zero came because
someone constructing notice that as you
dig a piece of earth out to make a hole
there's something an indication that
there that should have a name zero kind
of maybe came from that observation the
power of mathematics lies in the way its
language and symbols have allowed us to
manipulate the world but this was a
world that followed the rules of God and
the church by the 17th century a new
breed of intellectual was emerging not
afraid to challenge authority there was
one man who dared to question all of the
philosophical and scientific assumptions
that had gone before this was someone
who was trying to promote a new way of
thinking using reason experimentation
and observation this was the young
Frenchman called Rene Descartes it was
while in a restless sleep in 1619 that
Descartes experienced a series of dreams
that would change his life and
mathematics the first two could be
better described as nightmares
but the third dream the third dream was
intriguing as his eyes scanned the room
he saw books on the bedroom table it
appeared and then disappeared the opened
one book of poems and at random cost the
opening line of one which read what road
shall I kiss you in life then someone
appeared out of thin air and recited
another verse saying simply what is and
is not as with dreams it's all about the
interpretation you place upon them in
Descartes case the effect of these
dreams was profound
[Music]
he was convinced that the dreams were
pointing him in a single direction
bringing together the whole of human
knowledge by the means of reason he was
nothing if not ambitious but his genius
led to perhaps one of the greatest
advances ever in the field of
mathematics as with so many brilliant
ideas
it was deceptively simple let's say that
I'm meeting a friend for a coffee now
I'm standing at the end of ensley
gardens and they are somewhere over on
Gordon Street it's very easy for me to
work out how to get there all I need to
do is go on a map and check the route in
this case three streets down and one
alone
it's Sinan's like an incredibly simple
idea but actually it revolutionized
mathematics
he showed that a pair of numbers can
uniquely determine the position of a
point in space it sounds trivial but
this was just the start it gets more
interesting when you apply this idea to
curves
as this point moves around a circle its
coordinates change and we can write down
an equation that precisely and uniquely
characterizes this circle for the first
time shapes could be described by a
formula by uniting the language of
numbers and equations and symbols with
shapes Descartes was able to expand the
horizons of mathematics
thus laying the foundations for the
modern scientific world what Descartes
and the other trailblazers like him do
was to question accepted wisdom at the
time they thought differently and the
result was that they delivered
monumental breakthroughs for our
understanding of the universe
Descartes lived in a time when many
philosophers backed up their arguments
with appeals to God but Descartes
preferred to place his trust in the
power of human logic and maths he
believed all ideas should have their
foundations in experience and reason
rather than tradition and authority it
still feels like maths belongs to a
discovered world but after Descartes
it's a world that is increasingly devoid
of a divine influence and we started
this episode with just one question is
mathematics invented or discovered and
based on the evidence so far I'm leaning
quite heavily towards discovered because
it doesn't seem to me to be possible
there's something so all-encompassing
could be the product of the human mind
alone
next time I see how new mathematical
systems allowed Newton to create his
laws of gravity and even started
describing the existence of things we
didn't know whether all more evidence
our maths being a discovery now this
could be a coincidence but I'm forced to
think again when I confront one of the
strangest mathematical concepts there is
infinity and it makes the question of
where the math is invented or discovered
a lot more difficult to answer what
makes our world work the way that it
does explore more about the magic and
mystery of mathematics and how it
impacts our everyday life just go to BBC
code at UK forward slash maths and
follow the links to the Open University
and Hannah returns next Wednesday at
9:00 in the meantime she's with Adam
Rutherford investigating everyday
mysteries from pain thresholds to deja
vu the curious cases of Rutherford and
Frey on iPlayer Radio tomorrow here on
BBC four Brian Cox marveling at our
place in the great scheme of things
human universe a date
you